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Results tagged with heat-equation
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user 99469
0
votes
1
answer
98
views
Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-...
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\mathbb R^d …
- 825
4
votes
1
answer
765
views
How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-...
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
P_t …
- 825
2
votes
2
answers
149
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \...
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ is …
- 825
0
votes
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \...
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $d:=1, Z \sim \mathcal N (0, 1)$ and $\ell := p_1$. We write $M_1 \lesssim M_2$ if there exi …
- 825
1
vote
1
answer
120
views
Does Gaussian heat kernel ensure $\int_{\mathbb R^d} (1+|x|) \sqrt{\ell_{t_0} (x)} \, \mathr...
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\int_{\bR^d} (1+|x|) \sqrt{\ell (x)} \diff x < \i …
- 825
1
vote
1
answer
113
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) ...
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \mathbb R$ …
- 825
1
vote
1
answer
66
views
Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu,...
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, …
- 825
10
votes
0
answers
414
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_...
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \mat …
- 825
2
votes
0
answers
109
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{...
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R^ …
- 825
3
votes
1
answer
256
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|_{\ …
- 825
2
votes
1
answer
203
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \exp …
- 825